C*-dynamics and Equilibrium States
by
Shoichiro Sakai
Tohoku Gakuin University, Japan

Let {A, exp(tD)} be a C^*-dynamics with a unital C^*-algebra A and let f be a KMS state (=equilibrium state) for the dynamics at inverse temperature b. Let h be a selfadjoint element of A and consider a ^*-derivation D +Dih in A. Then there is a unique KMS state f[h] for the C^*-dynamics {A, exp(t(D+Dih))} at b such that the state f[h] is quasi-equivalent to the state f. Let H be the selfadjoint portion of A and for a positive number r, set K = {f[h] | h are all elements of H with || h || less than or equal to r}. Then one of the main results in the speaker's book [Operator Algebras in Dynamical Systems, Cambridge University Press, 1991] is the following theorem.

Theorem. K is relatively, weakly (namely, \sigma(A^*, A^**)) compact, where A^* (resp., A^**) is the dual (resp., bidual) Banach space of A. This theorem has many important applications in the bounded perturbation theory of C^*-dynamics.

The proof of this theorem in the book is rather complicated and so it is desirable to find a simpler proof. In this talk, the speaker presents a simpler proof and raises some problems related to this matter.

Date received: March 24, 1999